The fundamental importance of formal implication is brought out by the consideration that it is involved in all the rules of inference. This shows that we cannot hope wholly to define it in terms of material implication, but that some further element or elements must be involved. We may observe, however, that, in a particular inference, the rule according to which the inference proceeds is not required as a premiss. This point has been emphasized by Mr Bradley^{[37]}; it is closely connected with the principle of dropping a true premiss, being again a respect in which formalism breaks down. In order to apply a rule of inference, it is formally necessary to have a premiss asserting that the present case is an instance of the rule; we shall the need to affirm the rule by which we can go from the rule to an instance, and also affirm that here we have an instance of this rule, and so on into an endless process. The fact is, of course, that any implication warranted by a rule of inference does actually hold, and is not merely implied by the rule. This is simply an instance of the non-formal principle of dropping a true premiss: if our rule implies a certain implication, the rule may be dropped and the implication asserted. But it remains the case that the fact that our rule does imply the said implication, if introduced at all, must be simply perceived, and is not guaranteed by any formal deduction; and often it is just as easy, and consequently just as legitimate, to perceive immediately the implication in question as to perceive that it is implied by one or more of the rules of inference.(§ 45 ¶ 1)

To sum up our discussion of formal implication: a formal implication, we said, is the affirmation of *every* material implication of a certain class; and the class of material implications involved is, in simple cases, the class of all propositions in which a given fixed assertion, made concerning a certain subject or subjects, is affirmed to imply another given fixed assertion concerning the same subject or subjects. Where a formal implication holds, we agreed to regard it, wherever possible, as due to some relation between the assertions concerned. This theory raises many formidable logical problems, and requires, for its defence, a thorough analysis of the constituents of propositions. To this task we must now address ourselves.(§ 45 ¶ 2)

§ 45 n. 1. Logic, Book II, Part I, Chap. II (p. 227) ↩

The Principles of Mathematics was written by Bertrand Russell, and published in in 1903. It is now available in the Public Domain.